DOCSLIB.ORG

A Solution to the Arf-Kervaire Invariant Problem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Solution to the Arf-Kervaire Invariant Problem Proceedings of 17th G¨okova Geometry-Topology Conference pp. 21 – 63 A solution to the Arf-Kervaire invariant problem Michael A. Hill, Michael J. Hopkins and Douglas C. Ravenel Abstract. This paper gives the history and background of one of the oldest problems in algebraic topology, along with an outline of our solution to it. A rigorous account can be found in our preprint [HHR]. The third author has a website with numerous links to related papers and talks we have given on the subject since announcing our result in April, 2009. Drawing by Carolyn Snaith, 1981 Partially supported by DARPA and NSF. 21 HILL, HOPKINS and RAVENEL Contents 1. Background and history 24 1.1. Pontryagin’s early work on homotopy groups of spheres 24 1.2. Our main result 27 1.3. The manifold formulation 28 1.4. The unstable formulation 30 1.5. Questions raised by our theorem 32 2. Our strategy 33 2.1. Ingredients of the proof 33 2.2. The spectrum Ω 33 2.3. How we construct Ω 34 2.4. The norm 34 2.5. Fixed points and homotopy fixed points 35 2.6. RO(G)-graded homotopy 36 3. The slice filtration 37 3.1. Postnikov towers 37 3.2. Slice cells 38 3.3. The slice tower 38 3.4. The Slice Theorem 39 u (G) 3.5. Refining π∗ MU 40 3.6. The definition of ΩO 40 4. The Gap Theorem 41 4.1. Derivation from the Slice Theorem 41 4.2. An easy calculation 41 5. The Periodicity Theorem 44 5.1. Geometric fixed points 44 5.2. Some slice differentials 46 5.3. Some RO(G)-graded calculations 47 5.4. An even trickier RO(G)-graded calculation 47 5.5. The proof of the Periodicity Theorem 48 6. The Homotopy Fixed Point Theorem 49 7. The Detection Theorem 50 7.1. θj intheAdams-Novikovspectralsequence 50 7.2. Formal A-modules 52 7.3. The relation between MU (C8) and formal A-modules 53 7.4. DerivingtheDetectionTheoremfromLemma7.1 54 7.5. The proof of Lemma 7.1 54 8. The Slice and Reduction theorems 55 8.1. DerivingtheSliceTheoremfromtheReductionTheorem 56 8.2. The proof of the Reduction Theorem 57 References 61 22 A solution to the Arf-Kervaire invariant problem S Main Theorem. The Arf-Kervaire elements θj ∈ π2j+1−2 do not exist for j ≥ 7. S Here πk denotes the kth stable homotopy group of spheres, which will be defined shortly. The kth (for a positive integer k) homotopy group of the topological space X, denoted k by πk(X), is the set of continuous maps to X from the k-sphere S , up to continuous deformation. For technical reasons we require that each map send a specified point in Sk (called a base point) to a specified point x0 ∈ X. When X is path connected the choice of these two points is irrelevant, so it is usually omitted from the notation. When X is not path connected, we get different collections of maps depending on the path connected component of the base point. This set has a natural group structure, which is abelian for k > 1. The word nat- ural here means that a continuous base point preserving map f : X → Y induces a homomorphism f∗ : πkX → πkY , sometimes denoted by πkf. n It is known that the group πn+kS is independent of n for n>k. There is a homomor- n n+1 n+1 n+k+1 phism E : πn+kS → πn+k+1S defined as follows. S [respectively S ] can be obtained from Sn [Sn+k] by a double cone construction known as suspension. The cone over Sn is an (n+1)-dimensional ball, and gluing two such balls together along their com- mon boundary gives an (n+1)-dimensional sphere. A map f : Sn+k → Sn can be canoni- cally extended (by suspending both its source and target) to a map Ef : Sn+k+1 → Sn+1, and this leads to the suspension homomorphism E. The Freudenthal Suspension Theo- rem [Fre38], proved in 1938, says that it is onto for k = n and an isomorphism for n>k. n For this reason the group πn+kS is said to be stable when n>k, and it is denoted by S πk and called the stable k-stem. j+1 The Main Theorem above concerns the case k = 2 − 2. The θj in the theorem is a hypothetical element related to a geometric invariant of certain manifolds studied originally by Pontryagin starting in the 1930s, [Pon38], [Pon50] and [Pon55]. The problem came into its present form with a theorem of Browder [Bro69] published in 1969. There were several unsuccessful attempts to solve it in the 1970s. They were all aimed at proving the opposite of what we have proved, namely that all of the θj exist. The θj in the theorem is the name given to a hypothetical map between spheres for which the Arf-Kervaire invariant is nontrivial. Browder’s theorem says that such things can exist only in dimensions that are 2 less than a power of 2. Some homotopy theorists, most notably Mahowald, speculated about what would happen if θj existed for all j. They derived numerous consequences about homotopy groups of spheres. The possible nonexistence of the θj for large j was known as the Doomsday Hypothesis. After 1980, the problem faded into the background because it was thought to be too hard. In 2009, just a few weeks before we announced our theorem, Snaith published a book [Sna09] on the problem “to stem the tide of oblivion.” On the difficulty of the problem, he wrote 23 HILL, HOPKINS and RAVENEL In the light of ...the failure over fifty years to construct framed manifolds of Arf-Kervaire invariant one this might turn out to be a book about things which do not exist. This [is] why the quotations which preface each chapter contain a preponderance of utterances from the pen of Lewis Carroll. Our proof is two giant steps away from anything that was attempted in the 70s. We now know that the world of homotopy theory is very different from what they had envisioned then. 1. Background and history 1.1. Pontryagin’s early work on homotopy groups of spheres The Arf-Kervaire invariant problem has its origins in Pontryagin’s early work on a geometric approach to the homotopy groups of spheres, [Pon38], [Pon50] and [Pon55]. Pontryagin’s approach to maps f : Sn+k → Sn is to assume that f is smooth and that the base point y0 of the target is a regular value. (Any continuous f can be continuously −1 deformed to a map with this property.) This means that f (y0) is a closed smooth n+k n k-manifold M in S . Let D be the closure of an open ball around y0. If it is sufficiently small, then V n+k = f −1(Dn) in Sn+k is an (n + k)-manifold homeomorphic to M × Dn with boundary homeomorphic to M × Sn−1. It is also a tubular neighborhood of M k and comes equipped with a map p : V n+k → M k sending each point to the nearest point in M. For each x ∈ M, p−1(x) is homeomorphic to a closed n-ball Bn. The pair (p,f|V n+k) defines an explicit homeomorphism (p,f|V n+k) n+k k n V ≈ / M × D . This structure on M k is called a framing, and M is said to be framed in Rn+k. A choice n of basis of the tangent space at y0 ∈ S pulls back to a set of linearly independent normal vector fields on M ⊂ Rn+k. Conversely, suppose we have a closed sub-k-manifold M ⊂ Rn+k with a closed tubular neighborhood V and a homeomorphism h to M × Dn as above. This is called a framed sub-k-manifold of Rn+k. Some remarks are in order here. • The existence of a framing puts some restrictions on the topology of M. All of its characteristic classes must vanish. In particular it must be orientable. • A framing can be twisted by a map g : M → SO(n), where SO(n) denotes the group of orthogonal n × n matrices with determinant 1. Such matrices act on Dn in an obvious way. The twisted framing is the composite V h / M k × Dn / M k × Dn (m,x) / (m,g(m)(x)). We will say more about this later. 24 A solution to the Arf-Kervaire invariant problem • If we drop the assumption that M is framed, then the tubular neighborhood V is a (possibly nontrivial) disk bundle over M. The map M → y0 needs to be replaced by a map to the classifying space for such bundles, BO(n). This leads to unoriented bordism theory, which was analyzed by Thom in [Tho54]. Two helpful references for this material are the books by Milnor-Stasheff [MS74] and Stong [Sto68]. Pontryagin constructs a map P (M,h) : Sn+k → Sn as follows. We regard Sn+k as the one point compactification of Rn+k and Sn as the quotient Dn/∂Dn. This leads to a diagram p (V,∂V ) h / M × (Dn,∂Dn) 2 / (Dn,∂Dn) _ P (M,h) (Rn+k, Rn+k − intV ) / (Sn+k,Sn+k − intV ) / (Sn, {∞}) P (M,h) Sn+k − intV / {∞} The map P (M,h) is the extension of p2h obtained by sending the complement of V in Sn+k to the point at infinity in Sn.
Load more

Recommended publications
  • Obstructions to Approximating Maps of N-Manifolds Into R2n by Embeddings
    Topology and its Applications 123 (2002) 3–14 Obstructions to approximating maps of n-manifolds into R2n by embeddings P.M. Akhmetiev a,1,D.Repovšb,∗,A.B.Skopenkov c,1 a IZMIRAN, Moscow Region, Troitsk, Russia 142092 b Institute for Mathematics, Physics and Mechanics, University of Ljubljana, P.O. Box 2964, Ljubljana, Slovenia 1001 c Department of Mathematics, Kolmogorov College, 11 Kremenchugskaya, Moscow, Russia 121357 Received 11 November 1998; received in revised form 19 August 1999 Abstract We prove a criterion for approximability by embeddings in R2n of a general position map − f : K → R2n 1 from a closed n-manifold (for n 3). This approximability turns out to be equivalent to the property that f is a projected embedding, i.e., there is an embedding f¯: K → R2n such − that f = π ◦ f¯,whereπ : R2n → R2n 1 is the canonical projection. We prove that for n = 2, the obstruction modulo 2 to the existence of such a map f¯ is a product of Arf-invariants of certain quadratic forms. 2002 Elsevier Science B.V. All rights reserved. AMS classification: Primary 57R40; 57Q35, Secondary 54C25; 55S15; 57M20; 57R42 Keywords: Projected embedding; Approximability by embedding; Resolvable and nonresolvable triple point; Immersion; Arf-invariant 1. Introduction Throughout this paper we shall work in the smooth category. A map f : K → Rm is said to be approximable by embeddings if for each ε>0 there is an embedding φ : K → Rm, which is ε-close to f . This notion appeared in studies of embeddability of compacta in Euclidean spaces—for a recent survey see [15, §9] (see also [2], [8, §4], [14], [16, * Corresponding author.
    [Show full text]
  • The Kervaire Invariant in Homotopy Theory
    The Kervaire invariant in homotopy theory Mark Mahowald and Paul Goerss∗ June 20, 2011 Abstract In this note we discuss how the first author came upon the Kervaire invariant question while analyzing the image of the J-homomorphism in the EHP sequence. One of the central projects of algebraic topology is to calculate the homotopy classes of maps between two finite CW complexes. Even in the case of spheres – the smallest non-trivial CW complexes – this project has a long and rich history. Let Sn denote the n-sphere. If k < n, then all continuous maps Sk → Sn are null-homotopic, and if k = n, the homotopy class of a map Sn → Sn is detected by its degree. Even these basic facts require relatively deep results: if k = n = 1, we need covering space theory, and if n > 1, we need the Hurewicz theorem, which says that the first non-trivial homotopy group of a simply-connected space is isomorphic to the first non-vanishing homology group of positive degree. The classical proof of the Hurewicz theorem as found in, for example, [28] is quite delicate; more conceptual proofs use the Serre spectral sequence. n Let us write πiS for the ith homotopy group of the n-sphere; we may also n write πk+nS to emphasize that the complexity of the problem grows with k. n n ∼ Thus we have πn+kS = 0 if k < 0 and πnS = Z. Given the Hurewicz theorem and knowledge of the homology of Eilenberg-MacLane spaces it is relatively simple to compute that 0, n = 1; n ∼ πn+1S = Z, n = 2; Z/2Z, n ≥ 3.
    [Show full text]
  • The Arf and Sato Link Concordance Invariants
    transactions of the american mathematical society Volume 322, Number 2, December 1990 THE ARF AND SATO LINK CONCORDANCE INVARIANTS RACHEL STURM BEISS Abstract. The Kervaire-Arf invariant is a Z/2 valued concordance invari- ant of knots and proper links. The ß invariant (or Sato's invariant) is a Z valued concordance invariant of two component links of linking number zero discovered by J. Levine and studied by Sato, Cochran, and Daniel Ruberman. Cochran has found a sequence of invariants {/?,} associated with a two com- ponent link of linking number zero where each ßi is a Z valued concordance invariant and ß0 = ß . In this paper we demonstrate a formula for the Arf invariant of a two component link L = X U Y of linking number zero in terms of the ß invariant of the link: arf(X U Y) = arf(X) + arf(Y) + ß(X U Y) (mod 2). This leads to the result that the Arf invariant of a link of linking number zero is independent of the orientation of the link's components. We then estab- lish a formula for \ß\ in terms of the link's Alexander polynomial A(x, y) = (x- \)(y-\)f(x,y): \ß(L)\ = \f(\, 1)|. Finally we find a relationship between the ß{ invariants and linking numbers of lifts of X and y in a Z/2 cover of the compliment of X u Y . 1. Introduction The Kervaire-Arf invariant [KM, R] is a Z/2 valued concordance invariant of knots and proper links. The ß invariant (or Sato's invariant) is a Z valued concordance invariant of two component links of linking number zero discov- ered by Levine (unpublished) and studied by Sato [S], Cochran [C], and Daniel Ruberman.
    [Show full text]
  • The Arf-Kervaire Invariant Problem in Algebraic Topology: Introduction
    THE ARF-KERVAIRE INVARIANT PROBLEM IN ALGEBRAIC TOPOLOGY: INTRODUCTION MICHAEL A. HILL, MICHAEL J. HOPKINS, AND DOUGLAS C. RAVENEL ABSTRACT. This paper gives the history and background of one of the oldest problems in algebraic topology, along with a short summary of our solution to it and a description of some of the tools we use. More details of the proof are provided in our second paper in this volume, The Arf-Kervaire invariant problem in algebraic topology: Sketch of the proof. A rigorous account can be found in our preprint The non-existence of elements of Kervaire invariant one on the arXiv and on the third author’s home page. The latter also has numerous links to related papers and talks we have given on the subject since announcing our result in April, 2009. CONTENTS 1. Background and history 3 1.1. Pontryagin’s early work on homotopy groups of spheres 3 1.2. Our main result 8 1.3. The manifold formulation 8 1.4. The unstable formulation 12 1.5. Questions raised by our theorem 14 2. Our strategy 14 2.1. Ingredients of the proof 14 2.2. The spectrum Ω 15 2.3. How we construct Ω 15 3. Some classical algebraic topology. 15 3.1. Fibrations 15 3.2. Cofibrations 18 3.3. Eilenberg-Mac Lane spaces and cohomology operations 18 3.4. The Steenrod algebra. 19 3.5. Milnor’s formulation 20 3.6. Serre’s method of computing homotopy groups 21 3.7. The Adams spectral sequence 21 4. Spectra and equivariant spectra 23 4.1.
    [Show full text]
  • Durham Research Online
    Durham Research Online Deposited in DRO: 25 April 2019 Version of attached le: Published Version Peer-review status of attached le: Peer-reviewed Citation for published item: Cha, Jae Choon and Orr, Kent and Powell, Mark (2020) 'Whitney towers and abelian invariants of knots.', Mathematische Zeitschrift., 294 (1-2). pp. 519-553. Further information on publisher's website: https://doi.org/10.1007/s00209-019-02293-x Publisher's copyright statement: c The Author(s) 2019. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Additional information: Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full DRO policy for further details. Durham University Library, Stockton Road, Durham DH1 3LY, United Kingdom Tel : +44 (0)191 334 3042 | Fax : +44 (0)191 334 2971 https://dro.dur.ac.uk Mathematische Zeitschrift https://doi.org/10.1007/s00209-019-02293-x Mathematische Zeitschrift Whitney towers and abelian invariants of knots Jae Choon Cha1,2 · Kent E.
    [Show full text]
  • A Topologist's View of Symmetric and Quadratic
    1 A TOPOLOGIST'S VIEW OF SYMMETRIC AND QUADRATIC FORMS Andrew Ranicki (Edinburgh) http://www.maths.ed.ac.uk/eaar Patterson 60++, G¨ottingen,27 July 2009 2 The mathematical ancestors of S.J.Patterson Augustus Edward Hough Love Eidgenössische Technische Hochschule Zürich G. H. (Godfrey Harold) Hardy University of Cambridge Mary Lucy Cartwright University of Oxford (1930) Walter Kurt Hayman Alan Frank Beardon Samuel James Patterson University of Cambridge (1975) 3 The 35 students and 11 grandstudents of S.J.Patterson Schubert, Volcker (Vlotho) Do Stünkel, Matthias (Göttingen) Di Möhring, Leonhard (Hannover) Di,Do Bruns, Hans-Jürgen (Oldenburg?) Di Bauer, Friedrich Wolfgang (Frankfurt) Di,Do Hopf, Christof () Di Cromm, Oliver ( ) Di Klose, Joachim (Bonn) Do Talom, Fossi (Montreal) Do Kellner, Berndt (Göttingen) Di Martial Hille (St. Andrews) Do Matthews, Charles (Cambridge) Do (JWS Casels) Stratmann, Bernd O. (St. Andrews) Di,Do Falk, Kurt (Maynooth ) Di Kern, Thomas () M.Sc. (USA) Mirgel, Christa (Frankfurt?) Di Thirase, Jan (Göttingen) Di,Do Autenrieth, Michael (Hannover) Di, Do Karaschewski, Horst (Hamburg) Do Wellhausen, Gunther (Hannover) Di,Do Giovannopolous, Fotios (Göttingen) Do (ongoing) S.J.Patterson Mandouvalos, Nikolaos (Thessaloniki) Do Thiel, Björn (Göttingen(?)) Di,Do Louvel, Benoit (Lausanne) Di (Rennes), Do Wright, David (Oklahoma State) Do (B. Mazur) Widera, Manuela (Hannover) Di Krämer, Stefan (Göttingen) Di (Burmann) Hill, Richard (UC London) Do Monnerjahn, Thomas ( ) St.Ex. (Kriete) Propach, Ralf ( ) Di Beyerstedt, Bernd
    [Show full text]
  • Remarks on the Green-Schwarz Terms of Six-Dimensional Supergravity
    Remarks on the Green-Schwarz terms of six-dimensional supergravity theories Samuel Monnier1, Gregory W. Moore2 1 Section de Mathématiques, Université de Genève 2-4 rue du Lièvre, 1211 Genève 4, Switzerland [email protected] 2NHETC and Department of Physics and Astronomy Rutgers University Piscataway, NJ 08855, USA [email protected] Abstract We construct the Green-Schwarz terms of six-dimensional supergravity theories on space- times with non-trivial topology and gauge bundle. We prove the cancellation of all global gauge and gravitational anomalies for theories with gauge groups given by products of Upnq, SUpnq and Sppnq factors, as well as for E8. For other gauge groups, anomaly cancellation is equivalent to the triviality of a certain 7-dimensional spin topological field theory. We show in the case of a finite Abelian gauge group that there are residual global anomalies imposing constraints on the 6d supergravity. These constraints are compatible with the known F-theory models. Interest- ingly, our construction requires that the gravitational anomaly coefficient of the 6d supergravity theory is a characteristic element of the lattice of string charges, a fact true in six-dimensional F-theory compactifications but that until now was lacking a low-energy explanation. We also arXiv:1808.01334v2 [hep-th] 25 Nov 2018 discover a new anomaly coefficient associated with a torsion characteristic class in theories with a disconnected gauge group. 1 Contents 1 Introduction and summary 3 2 Preliminaries 8 2.1 Six-dimensional supergravity theories . ........... 8 2.2 Some facts about anomalies . 12 2.3 Anomalies of six-dimensional supergravities .
    [Show full text]
  • Higher-Order Intersections in Low-Dimensional Topology
    Higher-order intersections in SPECIAL FEATURE low-dimensional topology Jim Conanta, Rob Schneidermanb, and Peter Teichnerc,d,1 aDepartment of Mathematics, University of Tennessee, Knoxville, TN 37996-1300; bDepartment of Mathematics and Computer Science, Lehman College, City University of New York, 250 Bedford Park Boulevard West, Bronx, NY 10468; cDepartment of Mathematics, University of California, Berkeley, CA 94720-3840; and dMax Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany Edited by Robion C. Kirby, University of California, Berkeley, CA, and approved February 24, 2011 (received for review December 22, 2010) We show how to measure the failure of the Whitney move in dimension 4 by constructing higher-order intersection invariants W of Whitney towers built from iterated Whitney disks on immersed surfaces in 4-manifolds. For Whitney towers on immersed disks in the 4-ball, we identify some of these new invariants with – previously known link invariants such as Milnor, Sato Levine, and Fig. 1. (Left) A canceling pair of transverse intersections between two local Arf invariants. We also define higher-order Sato–Levine and Arf sheets of surfaces in a 3-dimensional slice of 4-space. The horizontal sheet invariants and show that these invariants detect the obstructions appears entirely in the “present,” and the red sheet appears as an arc that to framing a twisted Whitney tower. Together with Milnor invar- is assumed to extend into the “past” and the “future.” (Center) A Whitney iants, these higher-order invariants are shown to classify the exis- disk W pairing the intersections. (Right) A Whitney move guided by W tence of (twisted) Whitney towers of increasing order in the 4-ball.
    [Show full text]
  • Brown-Kervaire Invariants
    Brown-Kervaire Invariants Dissertation zur Erlangung des Grades ”Doktor der Naturwissenschaften” am Fachbereich Mathematik der Johannes Gutenberg-Universit¨at in Mainz Stephan Klaus geb. am 5.10.1963 in Saarbr¨ucken Mainz, Februar 1995 Contents Introduction 3 Conventions 8 1 Brown-Kervaire Invariants 9 1.1 The Construction ................................. 9 1.2 Some Properties .................................. 14 1.3 The Product Formula of Brown ......................... 16 1.4 Connection to the Adams Spectral Sequence .................. 23 Part I: Brown-Kervaire Invariants of Spin-Manifolds 27 2 Spin-Manifolds 28 2.1 Spin-Bordism ................................... 28 2.2 Brown-Kervaire Invariants and Generalized Kervaire Invariants ........ 35 2.3 Brown-Peterson-Kervaire Invariants ....................... 37 2.4 Generalized Wu Classes ............................. 40 3 The Ochanine k-Invariant 43 3.1 The Ochanine Signature Theorem and k-Invariant ............... 43 3.2 The Ochanine Elliptic Genus β ......................... 45 3.3 Analytic Interpretation .............................. 48 4 HP 2-BundlesandIntegralEllipticHomology 50 4.1 HP 2-Bundles ................................... 50 4.2 Integral Elliptic Homology ............................ 52 4.3 Secondary Operations and HP 2-Bundles .................... 54 5 AHomotopyTheoreticalProductFormula 56 5.1 Secondary Operations .............................. 56 5.2 A Product Formula ................................ 58 5.3 Application to HP 2-Bundles ........................... 62 6
    [Show full text]
  • [Math.RA] 24 Mar 2005 EEAIE R NAINSADREDUCED and INVARIANTS ARF GENERALIZED OE PRTOSI YLCHOMOLOGY CYCLIC in OPERATIONS POWER
    GENERALIZED ARF INVARIANTS AND REDUCED POWER OPERATIONS IN CYCLIC HOMOLOGY arXiv:math/0503538v1 [math.RA] 24 Mar 2005 GENERALIZED ARF INVARIANTS AND REDUCED POWER OPERATIONS IN CYCLIC HOMOLOGY een wetenschappelijke proeve op het gebied van de wiskunde en informatica PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Katholieke Universiteit te Nijmegen, volgens besluit van het college van decanen in het openbaar te verdedigen op donderdag 27 september 1990 des namiddags te 1.30 uur precies door PAULUS MARIA HUBERT WOLTERS geboren op 1 juli 1961 te Heerlen PROMOTOR: PROF. DR. J.H.M. STEENBRINK CO-PROMOTOR: DR. F.J.-B.J. CLAUWENS CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG Wolters, Paulus Maria Hubert Generalized Arf invariants and reduced power operations in cyclic homology / Paulus Maria Hubert Wolters. -[S.l. : s.n.]. - Ill. Proefschrift Nijmegen. - Met lit. opg. - Met samenvatting in het Nederlands. ISBN 90-9003548-6 SISO 513 UDC 512.66(043.3) Trefw.: homologische algebra. Contents Introduction vii List of some frequently used notations xi Chapter I Algebraic K-theory of quadratic forms and the Arf invariant 1 Collecting the relevant K- and L-theory 1 §2 The Arf invariant 16 § Chapter II New Invariants for L-groups 1 Extension of the anti-structure to the ring of formal power series 27 § s,h 2 Construction of the invariants ω1 and ω2 28 §3 Recognition of ωh 30 § 1 4 Computations on the invariant ω2 39 §5 Examples 51 § Chapter III Hochschild, cyclic and quaternionic homology 1 Definitions and notations 59 §2 Reduced power operations 63 §3 Morita invariance 76 §4 Generalized Arf invariants 79 § Chapter IV Applications to group rings 1 Quaternionic homology of group rings 81 §2 Managing Coker(1 + ϑ) 93 §3 Groups with two ends 101 §4 Υ for groups with two ends 105 § References 117 Samenvatting 119 Curriculum Vitae 120 v Introduction.
    [Show full text]
  • Quadratic Forms and the Birman-Craggs Homomorphisms by Dennis Johnson1
    transactions of the american mathematical society Volume 261, Number 1, September 1980 QUADRATIC FORMS AND THE BIRMAN-CRAGGS HOMOMORPHISMS BY DENNIS JOHNSON1 Abstract. Let "DH^be the mapping class group of a genus g orientable surface M, and íg the subgroup of those maps acting trivially on the homology group HX(M, Z). Birman and Craggs produced homomorphisms from 9 to Z2 via the Rochlin invariant and raised the question of enumerating them; in this paper we answer their question. It is shown that the homomorphisms are closely related to the quadratic forms on H¡(M, Z-¡) which induce the intersection form; in fact, they are in 1-1 correspondence with those quadratic forms of Arf invariant zero. Furthermore, the methods give a description of the quotient of 9g by the intersec- tion of the kernels of all these homomorphisms. It is a Z2-vector space isomorphic to a certain space of cubic polynomials over HX(M, Z¿). The dimension is then computed and found to be (2/) + (if). These results are also extended to the case of a surface with one boundary component, and in this situation the linear relations among the various homomorphisms are also determined. 1. Introduction. Let Mg be an orientable surface of genus g, <U\l its mapping class group and %gthe subgroup of those classes acting trivially on Hx(Mg, Z). In [BC], Birman and Craggs produced homomorphisms from jL to Z2 derived from the Rochlin invariant. Roughly, each / G Í is used to produce a homology 3-sphere, and its Rochlin invariant is the value of (one of) the homomorphism(s) on /.
    [Show full text]
  • The Kervaire Invariant
    The Kervaire invariant Paul VanKoughnett May 12, 2014 1 Introduction I was inspired to give this talk after seeing Jim Fowler give a talk at the Midwest Topology Seminar at IUPUI, in which he constructed a new class of non-triangulable manifolds. It came as something of a shock to see Fowler using characteristic classes, homotopy groups, and so on to study strange, pathological spaces out of geometry. For me, this was a much-needed reminder that the now-erudite techniques of homotopy theory arose from the study of honest geometric problems, and I decided to leran more of this theory. Kervaire's 1960 paper [5] turned out to be the perfect place to start. In it, Kervaire constructs the first known manifold admitting no smooth structure (in fact, it doesn't even admit any C1-differentiable structure!). This was fairly soon after Milnor had started studying exotic spheres, so these ideas were very much on topologists' minds. Moreover, in the course of his proof, he ends up using a lot of the great results of mid-century topology { the Pontryagin-Thom construction, the J-homomorphism, obstruction theory, and various results about the homotopy groups of spheres, among others. In particular, I had to relearn most of what I'd learned from Hatcher and Mosher-Tangora while preparing for this talk, which was a lot of fun. Kervaire limits his thought to 4-connected 10-dimensional manifolds M, whose cohomology is thus con- centrated in degrees 5 and 10. The cup product on the middle cohomology group is a skew-symmetric perfect pairing, and in mod 2 cohomology, Kervaire refines this to a quadratic form.
    [Show full text]